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No, there is no such graph. Suppose $\eta(G)=n$. Let $T_1, \dots, T_n$ be a collection of vertex disjoint trees in $G$ such that for all distinct $i,j \in [n]$, there is an edge $e(ij) \in E(G)$ betw …

Here is a follow-up to Richard Stanley's comment. In his Master's thesis Hunting for torus obstructions (University of Victoria, 2002), Chambers (together with his adviser Myrvold) studied the forbid …

Here is another answer that I just recently learned about that I find quite shocking. A graph is called $Y \Delta Y$ reducible if it can be reduced to isolated vertices by applying any of the followi …

A counterexample is the planar dual to the Tutte graph. Most of the credit goes to David Eppstein (see the comments below) Let $G$ be the dual to the Tutte graph. Then $G$ is a planar triangulation …

Yes, in my PhD thesis, we prove the stronger result that for any fixed signed graph $(G, \Sigma)$, there is a polynomial-time algorithm to test if an input signed graph contains a $(G, \Sigma)$-minor. …

This is false by classic results of Kostochka and Thomason. Indeed, the claim is false even if you replace 'minimum degree $t$' with '$t$-connected'. That is, if you define $\nu(t)$ to be the smalle …

More generally, it is a classic result (independently due to Kostochka and Thomason) that minimum degree $(\alpha+o(1))n \sqrt{\log n}$ suffices to force a $K_n$ minor, where $\alpha$ is an explicit c …

I have a copy of Sander's PhD thesis. Counting $K_6$, there are actually $76$ excluded minors for treewidth at most $4$ (found by computer) in the thesis, but it is unknown if this list is complete ( …

I'll just remark that the fact that every surface has a finite number of excluded minors (and also topological minors) does not require the full strength of the Graph Minors Theorem. Indeed, the proo …

The property that you are describing is called $2$-linked. More generally, we say that a graph is $k$-linked if it has at least $2k$ vertices and for all distinct vertices $s_1, \dots s_k, t_1, \dots …

There is no known necessary and sufficient condition like in Menger's theorem. However, there is a polynomial-time algorithm that decides if the paths exist. This is one of the main results of the Gr …

Yes. This is one of the results of the Matroid Minors Project of Geelen, Gerards and Whittle as part of their proof of Rota's Conjecture. In fact, they prove that for any finite abelian group $\Gamm …

Yes, this is true and follows from Wagner's theorem. Wagner's theorem asserts that every graph with no $K_5$ minor can be built from $0$-, $1$-, $2$-, and $3$-sums from planar graphs and a fixed $8$ …

No, it is not linear in the genus; it is at least exponential in $g$. See for example this answer by David Eppstein.

For your first question, Mader proved that all $K_6$-minor-free graphs (which includes all linklessly embeddable graphs) on $n$ vertices have at most $4n-10$ edges (thanks to David Eppstein for the re …